Subgroups — #28371905

Order 13952.

484 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812719	C₂	1	1	yes	maximal, characteristic
2	#20295506	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295126	C₂	1	2	yes	maximal, complemented
4	#20295124	C₂	1	2	yes	maximal, complemented
5	C₄×#13182019	C₂	1	2	yes	maximal, complemented
6	C₄×#13182014	C₂	1	2	yes	maximal, complemented
7	C₂×#16812632	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295027	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295025	C₂	1	1	yes	maximal, characteristic, complemented
10	#20294982	C₂	1	1	yes	maximal, characteristic, complemented
11	#520×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182029	C₂²	1	1	yes	characteristic
13	#16812719	C₄	1	8	yes	complemented
14	C₂×#13182019	C₂²	1	1	yes	characteristic
15	C₂×#13182017	C₂²	1	1	yes	characteristic
16	C₂×#13182016	C₂²	1	1	yes	characteristic
17	C₂×#13182014	C₂²	1	1	yes	characteristic
18	C₂×#13182013	C₂²	1	1	yes	characteristic
19	C₂²×#11724784	C₂²	1	1	yes	characteristic
20	C₄×Dic₂₁₈	C₂²	1	2	yes	—
21	C₄×D₄₃₆	C₂²	1	2	yes	—
22	#16812659	—	2	2	no	—
23	#16812657	—	2	1	no	—
24	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×D₁₀₉	—	2	1	no	—
26	#16812632	C₂²	1	2	yes	—
27	#16812632	C₂²	1	2	yes	—
28	#16812623	C₂²	1	2	yes	—
29	#16812622	C₂²	1	2	yes	—
30	#16812620	C₂²	1	2	yes	—
31	#16812619	C₂²	1	2	yes	—
32	#16812610	C₂²	1	2	yes	—
33	C₄×#11724784	C₂²	1	2	yes	—
34	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
36	C₄×D₄×C₁₀₉	—	2	2	no	—
37	C₄×D₄×C₁₀₉	C₂²	1	2	yes	—
38	#117×C₁₀₉	—	2	1	no	—
39	#117×C₁₀₉	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
41	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
42	#13182029	—	2	2	no	—
43	#13182029	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
45	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
47	C₂²×Dic₁₀₉	—	2	1	no	—
48	C₂×C₄×D₁₀₉	—	2	1	no	—
49	C₂×C₄×D₁₀₉	—	2	2	no	—
50	#13182019	C₂×C₄	1	4	yes	complemented
51	#13182017	C₂×C₄	1	4	yes	complemented
52	#13182016	C₂×C₄	1	4	yes	complemented
53	#13182014	C₂×C₄	1	4	yes	complemented
54	#13182013	—	2	2	no	—
55	C₂×#11724784	C₂×C₄	1	4	yes	complemented
56	C₂×#11724784	C₂³	1	2	yes	—
57	C₂×#11724784	C₂³	1	1	yes	characteristic
58	C₂×#11724784	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₁₀₉	—	2	1	no	—
60	C₄⋊Dic₁₀₉	C₂³	1	2	yes	—
61	#13182008	—	2	2	no	—
62	#13182006	—	2	2	no	—
63	C₄×Dic₁₀₉	—	2	2	no	—
64	#41×C₁₀₉	—	2	2	no	—
65	#41×C₁₀₉	C₂×C₄	1	4	yes	complemented
66	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₁₀₉	—	2	1	no	—
68	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₉	—	2	1	no	—
70	C₂²×C₄×C₁₀₉	—	2	1	no	—
71	C₂²×C₄×C₁₀₉	—	2	2	no	—
72	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₀₉	—	2	1	no	—
74	C₄⋊C₄×C₁₀₉	—	2	2	no	—
75	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
76	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	—
78	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
79	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
80	C₄²×C₁₀₉	—	2	2	no	—
81	C₄²×C₁₀₉	C₂³	1	2	yes	—
82	C₂³×C₁₀₉	—	2	1	no	—
83	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₁₀₉	—	2	1	no	—
85	Q₈×C₁₀₉	—	2	1	no	—
86	Q₈×C₁₀₉	C₂²×C₄	1	2	yes	—
87	D₄×C₁₀₉	—	2	4	no	—
88	D₄×C₁₀₉	—	2	1	no	—
89	D₄×C₁₀₉	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₀₉	—	2	1	no	—
91	C₂×C₄×C₁₀₉	—	2	4	no	—
92	C₂×C₄×C₁₀₉	—	2	2	no	—
93	C₂×C₄×C₁₀₉	—	8	2	no	—
94	C₂×C₄×C₁₀₉	—	8	1	no	—
95	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₀₉	#41	1	2	yes	—
100	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
101	C₂²⋊D₁₀₉	—	2	4	no	—
102	C₂×Dic₁₀₉	—	2	1	no	—
103	C₂×Dic₁₀₉	—	2	2	no	—
104	C₂×Dic₁₀₉	—	8	1	no	—
105	D₄₃₆	—	2	1	no	—
106	D₄₃₆	C₂²×C₄	1	2	yes	—
107	C₄×D₁₀₉	—	2	4	no	—
108	C₄×D₁₀₉	—	8	2	no	—
109	Dic₂₁₈	—	2	1	no	—
110	Dic₂₁₈	C₂²×C₄	1	2	yes	—
111	#11724784	—	2	2	no	—
112	#11724784	C₂²×C₄	1	2	yes	—
113	#11724784	C₂²×C₄	1	2	yes	—
114	C₂²×C₁₀₉	—	2	1	no	—
115	C₂²×C₁₀₉	—	2	1	no	—
116	C₂²×C₁₀₉	—	2	2	no	—
117	C₂²×C₁₀₉	—	4	1	no	—
118	C₂²×C₁₀₉	—	4	1	no	—
119	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
120	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₁₀₉	—	2	2	no	—
122	C₂×D₁₀₉	—	4	1	no	—
123	C₂×D₁₀₉	—	4	1	no	—
124	C₄×C₁₀₉	—	2	2	no	—
125	C₄×C₁₀₉	—	2	2	no	—
126	C₄×C₁₀₉	—	2	2	no	—
127	C₄×C₁₀₉	—	4	2	no	—
128	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
129	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
130	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
131	Dic₁₀₉	—	2	2	no	—
132	Dic₁₀₉	—	4	2	no	—
133	C₂×C₁₀₉	—	2	2	no	—
134	C₂×C₁₀₉	—	4	2	no	—
135	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂×C₁₀₉	#576	1	1	yes	characteristic
137	C₂×C₁₀₉	#578	1	1	yes	characteristic
138	D₁₀₉	—	4	2	no	—
139	#2951	—	109	1	no	maximal, Sylow
140	C₁₀₉	#2951	1	1	yes	characteristic, complemented, Sylow
141	C₂×#135	—	109	1	no	—
142	#520	—	109	1	no	—
143	#520	D₁₀₉	1	1	yes	characteristic, complemented
144	#440	—	109	2	no	—
145	#440	—	109	2	no	—
146	C₄×Q₁₆	—	109	2	no	—
147	C₄×D₈	—	109	2	no	—
148	C₂×#107	—	109	1	no	—
149	#417	—	109	1	no	—
150	#417	—	109	1	no	—
151	#405	—	109	1	no	—
152	C₂×#41	—	109	1	no	—
153	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
154	#135	—	109	8	no	—
155	C₂×Q₁₆	—	109	1	no	—
156	C₂×SD₁₆	—	109	1	no	—
157	C₂×SD₁₆	—	109	1	no	—
158	C₂×D₈	—	109	1	no	—
159	C₂×M₁₆	—	109	1	no	—
160	C₂²×C₈	—	109	1	no	—
161	C₄×Q₈	—	109	2	no	—
162	C₄×Q₈	C₂×D₁₀₉	1	2	yes	—
163	C₄×D₄	—	2	2	no	—
164	C₄×D₄	—	109	2	no	—
165	C₄×D₄	—	218	2	no	—
166	C₄×D₄	C₂×D₁₀₉	1	2	yes	—
167	#117	—	2	1	no	—
168	#117	—	218	1	no	—
169	#117	C₂×D₁₀₉	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	109	1	no	—
172	C₂×C₄⋊C₄	—	218	1	no	—
173	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
174	#107	—	109	2	no	—
175	#107	—	109	2	no	—
176	#103	—	109	2	no	—
177	#103	—	109	2	no	—
178	#102	—	109	2	no	—
179	#102	—	109	2	no	—
180	#97	—	109	2	no	—
181	C₄×C₈	—	109	2	no	—
182	#41	—	2	2	no	—
183	#41	—	109	4	no	—
184	#41	—	218	2	no	—
185	#41	C₄×D₁₀₉	1	4	yes	complemented
186	C₂×Q₈	—	109	1	no	—
187	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	109	1	no	—
190	C₂×D₄	—	218	1	no	—
191	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	218	1	no	—
196	C₂²×C₄	—	218	2	no	—
197	C₂²×C₄	—	218	1	no	—
198	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
199	Q₁₆	—	109	4	no	—
200	SD₁₆	—	109	4	no	—
201	SD₁₆	—	109	4	no	—
202	D₈	—	109	4	no	—
203	M₁₆	—	218	2	no	—
204	C₂×C₈	—	109	4	no	—
205	C₂×C₈	—	109	1	no	—
206	C₂×C₈	—	109	1	no	—
207	C₂×C₈	—	109	2	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	109	2	no	—
211	C₄⋊C₄	—	218	2	no	—
212	C₄⋊C₄	—	218	1	no	—
213	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
214	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
215	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	218	2	no	—
218	C₂²⋊C₄	C₂²⋊D₁₀₉	1	2	yes	—
219	C₄²	—	2	2	no	—
220	C₄²	—	218	2	no	—
221	C₄²	C₂²×D₁₀₉	1	2	yes	—
222	C₂³	—	2	1	no	—
223	C₂³	—	218	1	no	—
224	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	—	109	2	no	—
227	Q₈	—	218	1	no	—
228	Q₈	C₂×C₄×D₁₀₉	1	2	yes	—
229	D₄	—	2	4	no	—
230	D₄	—	2	1	no	—
231	D₄	—	109	2	no	—
232	D₄	—	218	4	no	—
233	D₄	—	218	1	no	—
234	D₄	C₂×C₄×D₁₀₉	1	2	yes	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	1	no	—
240	C₂×C₄	—	872	2	no	—
241	C₂×C₄	—	872	1	no	—
242	C₂×C₄	—	218	2	no	—
243	C₂×C₄	—	218	1	no	—
244	C₂×C₄	—	218	4	no	—
245	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
246	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
247	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
248	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
249	C₂×C₄	#13182029	1	2	yes	—
250	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
251	C₈	—	109	2	no	—
252	C₈	—	109	2	no	—
253	C₈	—	218	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	436	1	no	—
260	C₂²	—	436	1	no	—
261	C₂²	—	218	2	no	—
262	C₂²	C₄×C₂²⋊D₁₀₉	1	2	yes	—
263	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	436	2	no	—
269	C₄	—	218	2	no	—
270	C₄	C₄×C₂²⋊D₁₀₉	1	2	yes	—
271	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
272	C₄	C₂×#13182029	1	1	yes	characteristic
273	C₂	—	2	2	no	—
274	C₂	—	4	2	no	—
275	C₂	—	436	2	no	—
276	C₂	C₂×C₄×C₂²⋊D₁₀₉	1	1	yes	characteristic
277	C₂	#20295829	1	1	yes	characteristic
278	C₂	#20295841	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#520×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×#16812719	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#520	D₁₀₉
hypocenter	C₁₀₉	#2951
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #20295506	C₂	semidirect
3. #20295126	C₂	semidirect
4. #20295124	C₂	semidirect
5. C₄×#13182019	C₂	semidirect
6. C₄×#13182014	C₂	semidirect
7. C₂×#16812632	C₂	semidirect
8. #20295027	C₂	semidirect
9. #20295025	C₂	semidirect
10. #20294982	C₂	semidirect
11. #520×C₁₀₉	C₂	semidirect
13. #16812719	C₄	semidirect
43. #13182029	C₂×C₄	semidirect
50. #13182019	C₂×C₄	semidirect
51. #13182017	C₂×C₄	semidirect
52. #13182016	C₂×C₄	semidirect
53. #13182014	C₂×C₄	semidirect
55. C₂×#11724784	C₂×C₄	semidirect
65. #41×C₁₀₉	C₂×C₄	semidirect
140. C₁₀₉	#2951	semidirect
143. #520	D₁₀₉	semidirect
185. #41	C₄×D₁₀₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16812719	2
#20295506	2
#20295126	2
#20295124	2
C₄×#13182019	2
C₄×#13182014	2
C₂×#16812632	2
#20295027	2
#20295025	2
#20294982	2
#520×C₁₀₉	2
#2951	109